A mixing tree-valued process arising under neutral evolution with recombination

TL;DR

This paper introduces a new tree-valued process modeling genealogies along a genome under recombination, demonstrating convergence and mixing properties as population size grows large.

Contribution

It develops a limiting tree-valued process for genealogies across the genome under recombination, extending classical coalescent models.

Findings

Convergence of the genealogical process to a cadlag tree-valued process.

Analysis of mixing properties for distant loci.

Extension of Kingman's coalescent to multiple loci with recombination.

Abstract

The genealogy at a single locus of a constant size $N$ population in equilibrium is given by the well-known Kingman's coalescent. When considering multiple loci under recombination, the ancestral recombination graph encodes the genealogies at all loci in one graph. For a continuous genome $G$, we study the tree-valued process $(T^N_u)_{u\in G}$ of genealogies along the genome in the limit $N\to\infty$. Encoding trees as metric measure spaces, we show convergence to a tree-valued process with cadlag paths. In addition, we study mixing properties of the resulting process for loci which are far apart.